>. B fc*J 

HE EARTH 

AND ITS CHIEF MOTIONS, 



THE TANGENT INDEX, 



JOHN HAYWOOD, 

Professor of Mathematics, Otlerbein University, 
Westerville, Ohio. 



DAYTON, OHIO: 
Press of U. B. Publishing House. 

1888. 



THE EARTH 



AND ITS CHIEF MOTIONS, 



THE TANGENT INDEX. 



. 






BY 

JOHN HAYWOOD, 

Professor of Mathematics, Otterbein University, 
Westerville, Ohio. 



DAYTON, OHIO: 

Press of U. B. Publishing House. 

1888. 



Copyright, 1888, 
By John Haywood. 



3& 






The Earth and Its Chief Motions. 



CHAPTER I. 

OUR WORLD. 

The object of these lines is to furnish some "help to those who 
are studying the elements of astronomy, by assisting them to com- 
prehend the most important of our astronomical relations. 

We find ourselves inhabitants of a world which to our senses is 
infinite, and fixed in its place. When we attempt to explore it, we 
find ourselves held within certain limits. Some of these limits — 
seas, deserts, wildernesses — human ingenuity and perseverance 
enable the explorer to break through Other limits we know can 
never be passed over. Still others are doubtful. Thus we never 
expect to penetrate deep into the earth. We cannot rise much 
above its surface. Travelers ascending mountains, or navigating 
the air in balloons, imperil their lives, and not infrequently lose 
them. At most, the height to be attained in either of these ways is 
small. 

Again, the frozen zones seem to present impassable barriers to 
the daring navigators who have attempted to pass them ; and though 
many lives have been lost, and much treasure has been expended in 
the attempts to explore these regions, they remain, to a great extent, 
unknown. Hence our knowledge of our world is imperfect, and 
doubtless there will always be more of it unknown than known. 

With respect to other worlds, they are more inaccessible than the 
frozen zones or the interior of the earth. A great gulf, which no 
man can pass over, separates us from them. It might well seem 
that such distant and inaccessible worlds were not proper objects of 
investigation. On the contrary, this very study, astronomy, has 

3 



THE EARTH AND ITS CHIEF MOTIONS. 



helped us in carrying forward the study of our own world. With- 
out astronomy, science and civilization would necessarily be much 
more imperfect than they are. 

The science of astronomy is possible to us because we have the 
power of vision, and what is of more importance, the power of 
reasoning, by which, on the comparison of known truths, we deduce 
with certainty various connected truths. The certainty of these 
deductions is so great that we unhesitatingly correct our erroneous 
ideas derived from the more imperfect testimony of the senses. 
Thus the traveler in the desert sees what seems to be sheets of water, 
green herbage, trees, and buildings, but judges the appearances an 
illusion. Thus when we are on the water in a vessel, we cannot at 
once determine whether the motion we see is a motion of our ves- 
sel or of other vessels or objects near by. Sometimes when we are 
on a railroad train entering or leaving the station, we notice an illu- 
sion of the same kind. But the most complete and ineradicable 
illusion in our experience is exposed and corrected in the study of 
astronomy ; an illusion that is absolutely universal, and, do what we 
will, adheres to us so tenaciously that we can only correct it in 
thought, in judgment — the illusion that this world is at rest and 
fixed in space ; and that up and down are invariable directions in 
space. So deep-seated and unchangeable is the conviction derived 
from these appearances that astronomers do not try to change the 
language ordinarily used in speaking of the various astronomical 
phenomena, the rising and setting of the sun, moon, and stars, etc. 



THE EARTH AND ITS CHIEF MOTIONS. 



CHAPTER II. 

THE EARTH A SPHERE. 

To the eye, the earth presents the appearance of a boundless 
plain, diversified with mountains, hills, and valleys, plains and seas. 
In our experience, all objects rest upon something. If the support 
upon which an object rests be taken away, the object falls, descends 
toward the ground until it finds a support, either on the ground itself 
or upon something which rests upon the ground. This universal 
experience leads us naturally, when discussing the form of the earth, 
to inquire what supports the earth, which itself serves as the sup- 
port of all things belonging to it? It requires careful investigations 
to make certain, or to correct, our impressions and natural convic- 
tions on these points. 

The proofs of the spherical form of the earth are sufficiently 
numerous and convincing when we bring our reasoning powers to 
bear upon them. The manner of the disappearance of a ship sail- 
ing away from us is to the purpose. The body of the ship disap- 
pears below the surface of the water, while the upper parts, the 
masts and sails, are still visible ; then they, too, disappear below the 
water-line. In like manner, when a ship is coming toward us, we 
see the top-sails first, then the lower sails and the hull successively 
heave in sight. 

The shadow of the earth upon the moon when the latter is in 
eclipse also exhibits the same truth. We reason in this case, and 
in the preceding cases, that the appearances presented, (the phe- 
nomena), compel us to conclude that the earth is spherical. Other 
considerations, which we will not stop to present, lead us to the 
same conclusion. 

Now, admitting the sphericity of the earth, what do the words 
up and down mean ? Travelers in all parts of the world find the 
appearances of up and down the same as we. We may suppose a 
traveler on the opposite side of this globe — the earth — from us, 



6 THE EARTH AND ITS CHIEF MOTIONS. 

and may conceive a straight line, a diameter, passed through the 
center of the earth from our place to the opposite place. Now, evi- 
dently, down, to this traveler will be in exactly the opposite direc- 
tion in space to that indicated by down with us. We conclude, then, 
that the terms down and up relate to the earth, that down means 
toward the earth, toward its center, and up means away from the 
earth. 

Also, when we see unsupported objects fall to the ground, where- 
ever on the earth the observation is made, we conclude that by fall- 
ing we mean really a moving toward the earth ; that bodies are 
affected by some force which we call gravity, which urges them 
toward the earth. We may carry our conclusions further. If we 
could in some way place ourselves somewhere in space, away from 
this world and from any other worlds, m we may conclude that there 
would be no down nor up, that there would be no falling. Dwellers 
upon the earth, travelers wherever they are, find the earth, with all 
that belongs to it, separated from all other worlds and objects by 
an inconceivable distance. The heavens seem as distant from us 
in one part of the earth as in another. They seem above us in one 
part the same as in another. Indeed, we ourselves can here at 
home make these observations for ourselves, since we are carried 
around the earth in its diurnal motion, as will be more fully shown 
hereafter ; and we see a different heaven above us by night and by 
day. Therefore the earth is itself in the situation supposed of a 
body out in space, and so far separated from other worlds that it 
is so little affected by them that we may, for our present discussion, 
consider it quite unaffected; therefore the earth needs no founda- 
tion — that is, it is so far removed from other worlds that it keeps 
its place without any support. 



THE EARTH AND ITS CHIEF MOTIONS. 



CHAPTER III. 

THE EARTH A SPHERE.— Continued. 

It is therefore correct to say that the earth is but little affected by 
other worlds. It is, in fact, urged both toward the moon and toward 
the sun ; and, indeed, toward every member of the solar system, if 
not toward every star in the sky. But the intensity of gravity toward 
these distant bodies, even the nearest of them — the moon — is feeble, 
comparatively. This can be readily shown. A mass of any mater- 
ial, lead for instance, weighing 3,600 pounds on the earth, will 
weigh, at the distance of the moon — sixty times as far from the cen- 
ter of the earth — only one pound ; that is, the same muscular force 
exerted in holding up one pound here on the earth's surface, would, 
at the distance of the moon, hold up the mass of lead above men- 
tioned, so far as it is affected by terrestial gravity. 

It will be noticed that the term, weight, used in this connection, 
is ambiguous. Thus we have weighing machines in which the ma- 
terial to be weighed is balanced by counterpoises ; and we have ma- 
chines in which the weight is held up by steel springs, and the 
amount of weight is measured by the compression of the springs. 
If the mass of lead were weighed by a machine of the former kind, 
it would weigh as much at the distance of the moon as on the earth's 
surface, since the gravity of the counterpoises would be affected by 
the distance in the same proportion as that of the mass to be 
weighed ; but if the mass of lead were weighed by a spring balance 
at the distance of the moon, it would show a weight of one pound 
only. The increased distance, which causes a diminution of the in- 
tensity of gravity, of course does not affect the elasticity of the 
spring. 

Again, a body, as a ball of lead, near the earth, will fall from a 
state of rest by gravity, in one second, sixteen and one-twelfth feet 
— that is, by terrestial gravity of the intensity we experience on the 
earth ; but at the distance of the moon it would, under the same con- 
ditions, fall only 0.05 of an inch. 



O THE EARTH AND ITS CHIEF MOTIONS. 

Again, it is known that solar gravity is more than 300,000 times 
greater than terrestial gravity under the same conditions of distance. 
A piece of lead weighing a pound on the earth, would, at the sun's 
surface, weigh twenty-seven and a half pounds in a spring balance. 
It would fall from rest by solar gravity near the sun's surface, 440 
feet in one second. But at the earth's distance from the sun, the 
solar gravity is so enfeebled that a mass weighing on the earth one 
ton, would, by solar gravity, at that distance, weigh by a spring bal- 
ance only one pound ; and it would fall toward the sun from a state 
of rest only one-tenth of an inch in one second. 

It will be seen from these statements that the proposition that the 
earth is in the situation of a body removed from the influence of the 
gravity of other worlds is substantially proved. The important facl; 
that this enfeebled solar gravity is counterbalanced by centrifugal 
force generated by the revolution of the earth about the sun, is con- 
sidered further along. 

According to the statements made above, it will be readily seen 
that the moon is also a world, in some sort connected with the 
earth, but so remote that terrestial gravity is only sufficient to keep 
it from abandoning the earth as it moves on its course ; and we see 
here, also, the case of a world holding its place in space without 
support, and needing none. 



THE EARTH AND ITS CHIEF .MOTION'S. 



CHAPTER IV. 

ROTATION OF THE EARTH UPON ITS AXIS. 

Having found the earth a sphere, and holding its place in space 
and unsupported, and needing no support, it is much easier to see 
that it may turn on its axR There are many observed facts which 
prove the diurnal motion of the earth. It will be sufficient to pre- 
sent two of these most easily comprehended, and leading most di- 
rectly to the conclusion. 

The whole heavens seem to revolve around the earth every day. 
Every star, the sun and moon, are carried about the earth much as 
though they were attached to the inner surface of an immense hol- 
low sphere which surrounds the earth, and which revolves upon its 
axis. This is the appearance the heavens present, and this was the 
first thought of the ancient astronomers. But we now know that 
these different celestial bodies are not attached to the surface of a 
sphere ; that they are very unequally distant from us ; that all of 
them are at such great distances that it is inconceivable that they 
should revolve about the earth in one day. 

Another fact, not so obvious, is that the equatorial radius of the 
earth is about thirteen miles longer than the polar radius. This has 
been fully established by surveys made in different countries by the 
most accomplished engineers, and with the best surveying apparatus 
modern science and art have been able to furnish. Admitting this 
fact, we are compelled to admit the rotation of the earth, since if the 
waters of the equatorial seas were not held to the equatorial parts of 
the earth by some force to counteract gravity, they would flow to 
the north and south by the force of gravity, and form seas about the 
poles deep enough to bring the earth to the form of a sphere, ex- 
cept the enormous land elevations along the equator formed by the 
retreat of the waters. The centrifugal force generated by the 
rotation of the earth upon its axis just supplies this needed force ; 
and we conclude the form of the earth to be slightly spheroidal, 



IO THE EARTH AND ITS CHIEF MOTIONS. 

being the form required to bring the force of gravity and the 
centrifugal force on the different parts of the earth's surface to an 
equilibrium. 

Reasoning on these facts, we conclude that the earth turns on its 
axis every day, thus bringing about the succession of day and 
night. 



THK RARTH AND ITS CHIEF MOTIONS 



CHAPTER V. 

THE ANNUAL MOTION OF THE EARTH. 

Besides the vicissitudes of day and night, which we ascribe to the 
revolution of the earth on its axis, we find a continual change of 
seasons. This the earth's diurnal motion does not account for. 
But by careful watching through the year, and year after year, we 
see that the sun moves to the north and back to the south regularly 
each year ; that when the sun is far north it is summer, and when it 
is far south it is winter in our northern hemisphere. To those liv- 
ing in the southern hemisphere the seasons are just opposite ; so 
that summer in the one hemisphere is at the time it is winter in the 
other. The change of seasons, therefore, we find is caused by the 
change of the sun's place to the north or south. 

It is found convenient to refer the sun's place in this respect to the 
equator. Thus, to one living on the equator, the sun is half the 
year north of him, and the other half of the year south ; and twice 
in the year, March 21st and September 22d, it is on the equator — 
that is, on those days, the sun, in its apparent diurnal motion, rises 
exactly in the east, at noon is exactly over head, and sets exactly in 
the west — in other words, the sun's apparent path on these days 
coincides with the plane of the earth's equator. This path is called 
the celestial equator. It may also be conceived as a great circle of 
the heavens marked on the sky by an equa f orial radius of the earth 
prolonged to the sky. Thus if this radius were a pencil, and the 
sky were a solid surface, there would be traced on the spherical sur- 
face, as the earth turned on its axis, a great circle of the celestial 
sphere, the celestial equator or equinoctial. Although there is not 
in fact any such visible circle on the sky as described, yet it is con- 
ceived of, and the sun's place from day to day is determined with 
reference to it. The distance of the sun from this circle at any time 
is called its declination. It is estimated in circular measure -that is, 
in degrees, etc. It will be noticed that declination has much the 



12 TIHC EARTH AND ITS CHIEF MOTIONS. 

same meaning as latitude, used to indicate the situation of a place 
on the earth ; also, it may be here stated that the places of the other 
celestial bodies, the moon, the planets, and the stars, are referred to 
the equinoctial ; the word declination being used still with the same 
meaning ; namely, the distance of a "body north or south of the equ- 
inoctial." 

While this change of the sun's declination is taking place, we 
shall, if watchful, observe also a change in our visible heavens. 
Many stars and groups of stars (constellations) which we noticed 
in March, are not visible in September ; or, if visible, are in a differ- 
ent part of the sky. Noticing more closely, we see a constant shift- 
ing of the constellations to the west, in a constant, endless proces- 
sion, so that in a year's time they are seen returned to the position 
occupied at the beginning of the year. 

We may reason on these phenomena as we did in the matter of 
the diurnal motion. Instead of supposing the stars attached to the 
inner surface of an immense hollow sphere, we say they are at differ- 
ent distances ; and in any case they are too distant to move through 
such a circuit in one year ; therefore, we ascribe the phenomena to 
the motion of the sun, a much nearer body, eastward in a path about 
the earth. This path careful observation shows to be a great circle 
of the heavens called the ecliptic. It intersects the equinoctial at 
two opposite points of the heavens called the equinoxes. The sun 
is at the one, the vernal equinox, on March 21st, and is ttfen going 
northward in declination. It is at the other equinox, the autumnal, 
about September 22d ; at that time going south. On the day about 
midway between these two dates, June 21st, the sun is at the point 
of the ecliptic having the greatest northern declination, about twen- 
ty-three and a half degrees. This point is called the summer sol- 
stice. Also, midway between the autumnal equinox and the vernal, 
the sun reaches its greatest southern declination, twenty-three and a 
half degrees. This point is called the winter solstice. The sun is 
at this point of the ecliptic about December 21st. This angular 
number, twenty-three and one half degrees, measures the angle in- 
cluded between the two circles, the equator and the ecliptic. It is 
called the obliquity of the ecliptic. 

The circumference of the ecliptic is divided into twelve parts of 



THE EARTH AND ITS CHIEF MOTIONS. [3 

thirty degrees each, called signs. These signs have names given to 
them in the early history of astronomy. Thus, beginning at the 
vernal equinox, the first division eastward is called Aries, the sec- 
ond Taurus, etc. These names are found in all almanacs. The two 
equinoxes, therefore, are at the beginning of Aries and Libra ; the 
two solstices at the beginning of Cancer and Capricorn. 



14 THK EARTH AND ITS CHIEF MOTIONS. 



CHAPTER VI. 

THE ANNUAL MOTION OF THE EARTH.— Continued. 

The last two names above, Cancer and Capricorn, are also given 
to the two circles conceived as drawn on the earth as boundaries of 
the torrid zone. The significance of these two names is readily 
seen. Thus on June 21st, when the sun is at the summer sols' ice, 
and having a declination of twenty-three and a half degrees north, 
it is directly over a point on the earth having a latitude of twenty- 
three and a half degrees north ; and to a person on any point on the 
earth having this latitude, the sun at noon would be directly over- 
head. A circle, therefore, drawn around the earth parallel to the 
equator, and at a latitude of twenty-three and a half degrees north, 
passes through all the points on the earth at which the sun is verti- 
cal at noon on June 21st, when it is farthest north. This limit is 
taken as the northern boundary of the torrid zone. The circle is 
called tropic because the sun now stops (solstice) going north, and 
turns (tropic) southward. 

In the same way we see why the southern boundary of the torrid 
zone is called the Tropic of Capricorn. The sun, on December 
2 1 st, as at the winter solstice, and in the course of its diurnal motion 
that day is directly overhead at noon to all points of the earth that 
are situated on a parallel to the equator, at a distance of twenty- 
three and a half degrees south of it. At this time the sun begins to 
return toward the north. 

The two frigid zones are bounded by the two polar circles. These 
are small circles drawn around the poles, twenty-three and a half 
degrees from the poles. The reason for thus bounding these 
zones will be seen from the following considerations : When the 
sun is at the summer solstice, twenty-three and a half degrees north 
of the equinoctial, it will shine over all the north polar region to the 
distance of twenty-three and a half degrees, from the pole, and it 
will not set on that day to any part of this limited portion of the 



THE EARTH AND ITS CHIEF motions. 15 

earth's surface. To one standing at the pole, the sun would be seen 
to be twenty-three and a half degrees above the horizon, and to 
move around the horizon, keeping the same height above it, and 
making a complete circuit in twenty-four hours. In fact, it has been 
constantly in sight since March 21st, at which date it came up to the 
horizon, or rose. It came up slowly, higher and higher, still mak- 
ing its complete circuit every twenty four hours, till June 21st, or the 
summer solstice. Then it begins slowly to get lower, but all the 
time continuing its daily circuit, till September 22d, when it sets, to 
remain out of sight while the sun is passing from the autumnal equi- 
nox to the winter solstice, and on to the vernal equinox. Thus, at 
the pole, there is but one day and one night in a year, each six 
months long. At the south pole the appearances are the same, but 
reversed — that is, while it is day at one pole, it is night at the other. 

If we change our station from the pole, and move down toward 
the polar circle, the phenomena of day and night change gradually 
to what we ordinarily see here in the temperate zone — that is, there 
will be, beginning at the vernal equinox, a rising and setting of the 
sun, the days growing longer as the season advances ; then the sun 
will just skim the northern horizon at what might be called mid- 
night ; then the sun will remain continually above the horizon for 
days, or weeks, or months, as we are nearer the pole, until the 
summer solstice, and the same length of time after the solstice, 
when the sun will, at the hour of the day corresponding to midnight, 
just touch the northern horizon. Then, as the sun advances to the 
autumnal equinox, the succession of day and night is resumed, the 
days becoming shorter and the nigh's longer, until sometime after 
the autumnal equinox, the sun will, at noon, just come up to the 
southern horizon. Then it ceases to rise, and the night continues 
unbroken for a length of time about equal to the time the day con- 
tinued unbroken in the opposite part of the year — that is, till some 
time after the winter solstice, when the sun is again seen in the 
southern horizon. Then the succession of day and night begins 
again, the days at first very short, then longer, till the vernal equi- 
nox, when the days and nights are equal. The days from this time 
go on lengthening, as before described. 

But on the polar circle, at the time of the summer solstice, the 



l6 THE EARTH AND ITS CHIEF MOTIONS. 

sun just comes down to the northern horizon at midnight ; and then 
without going down out of sight, begins, in its diurnal motion, to 
rise as it moves around to the south where, at noon, it is forty-seven 
degrees high. Thus at all places on the polar circle, during a whole 
day at the summer solstice, the sun does not set, and for a whole 
day at the winter solstice the sun does not rise. This circle, there- 
fore, forms the boundary of the frigid zone — the north frigid zone 
about the north pole, and the south frigid zone about the south 
pole. 

The two temperate zones are bounded on the side toward the 
equator by the tropics ; and by the polar circles on the side toward 
the poles. 

This discussion shows how a knowledge of astronomy helps to 
make the facls of geography plain. 



THE EARTH AND ITS CHIEF .MOTIONS. 1 7 



CHAPTER VII. 

THE ANNUAL MOTION OF THIv IvARTH.— Continued. 

The phenomena of the change of the seasons are accounted for, 
as we have before seen, by an annual motion of the sun about the 
earth in a great circle, the ecliptic, inclined to the equinoctial twenty- 
three and a half degrees. This motion, therefore, we call the sun's 
apparent annual motion. But astronomy teaches that the earth 
moves annually about the sun. Let us see on what ground. The 
question is more complex than that relating to the diurnal motion of 
the earth. An experiment in physics will help to an understanding 
of the subject. Let us take two stones, two pieces of brick, or bet- 
ter, t \ o balls such as are used in playing ball, or better still, two 
leaden balls of suitable weight. Fasten them to the ends of a light 
rod of convenient length. Now we can support the rod on the edge 
of a knife in such a way that the two balls — we will call them bodies 
— will balance. If the two bodies are equal, the support will be un- 
der the middle point of the rod. If they are unequal, the support 
will be nearer the heavier body. In either case, the point at which 
they, balance is called their common centre of gravity. Now, tie a 
string to the rod at this point, and suspend the apparatus at a con- 
venient height from a nail or hook in the ceiling of the room. Then 
set the bodies to whirling about each other. It will be seen at once 
that each of them moves in a circle, and that the common centre of 
gravity is the common centre of both circles. 

Another way to exhibit this law of revolution, and in some re- 
spects a better way, is to tie the two bodies to the ends of a string 
of sufficient length, and suspend them side by side from a hook in 
the ceiling. Now twist the strings together evenly and to such a de- 
gree that the effort to untwist when the bodies are set free will cause 
a fair velocity of revolution. Then the two bodies will swing apart 
more and more widely as the swiftness of revolution increases, 



iS THE EARTH AND ITS CHIEF MOTIONS. 

while at the same time they are drawn toward each other by the 
action of the strings and their gravity toward the earth. Now, what- 
ever be the velocity of revolution, and their distance apart, it will be 
seen that they revolve about their common centre of gravity. This 
will be the ca^e whether the bodies are equal or unequal ; and, fur- 
ther, it will evidently be true whatever be their size, and if they are 
urged toward each other by their mutual gravity. Let, then, the 
two bodies be two worlds, situated in space at a suitable distance 
from each other and from all other worlds, so that they can move 
freely and not be swayed by any force but the mutual gravity which 
urges the two toward each other ; and let them be made to revolve 
with sufficient velocity to generate a centrifugal force great enough 
to counterbalance their mutual gravity ; then will the two worlds 
revolve about their common centre of gravity. 

If we should observe this motion from a position on one of these 
worlds, it would appear to us that our world was stationary, and 
that the other world is revolving around our world. But we should 
know from the principle proved above, that in fact, they were re- 
volving around their common centre of gravity. If these two 
worlds were of equal weight (mass), the centre would be midway 
between them. But if one had a very much greater mass than the 
other, it would be proper to say that the smaller world revolves 
about the larger, rather than the larger revolves about the smaller. 
But the facl is fully established that the mass of the sun is more 
than 300,000 times larger than the mass of the earth, as stated here- 
tofore ; therefore we say the earth revolves around the sun. On ac- 
count of this enormous inequality in the mass of the sun and earth, 
we could not see any motion in the sun if we were ever so favorably 
situated to see it. If the mass of the earth were very much greater 
than it is, if all the planets were joined to the earth, — some of them 
are several hundred times greater than the earth, — still we should 
see this one great planet revolving around the sun, and still the 
sun's motion would be too inconsiderable to be noticed ; so wonder- 
fully large a world is our sun. 

NOTE A. 

It is known that the sun has a motion of rotation on its axis in 
about twenty-five and one third days, and this motion is readily ob- 



THE KARTH AND ITS CHIEF MOTIONS. ]«i 

served. This is not considered in the discussion ; only the motion 
of displacement need be considered, and that is too little to be 
noticed, as stated above. 

NOTE B. 

In discussing the motion of two bodies about their common centre 
of gravity, the supposition was made that the bodies be removed 
from the influence of other bodies. Also that they be placed at a 
suitable distance from each other. The first supposition cannot be 
true of any two bodies in the solar system ; as, for instance, the Sun 
and Earth. Thus the earth» gravitates towards the other planets ; 
the intensity of the force being different with die varying distance 
of the disturbing body. But its greatest value is small compared 
with the solar gravity. Yet the effect is great enough to be noticed 
and taken account of by the astronomers ; and in preparing an 
ephemeris of a planet, as the earth, the disturbance caused by each 
of the other planets is computed and allowance is made. These 
disturbances are called perturbations. 

With regard to the distance of two bodies which are suppoed to 
constitute a system, it may be stated, that theoretically, two mate- 
rial points, that is things having gravity but not sensible magnitude, 
may form a system and perform their revolutions about each other 
at any distance however small. But bodies, having magnitude, need 
to be at a suitable distance from each other. It is obvious that the 
two bodies at their nearest approach must be at a distance greater 
than the sum of the radii of the two spheres. If the nearer parts 
were to collide, the system would of course collapse. But if the 
bodies approached closely without colliding, the unequal gravity of 
the nearer and remoter parts of the bodies would cause perturbations 
of sufficient magnitude perhaps to bring the system to an end. This 
is clearly the case when the bodies are two worlds. Thus the moon, 
as distant as it is, acts so unequally upon the nearer and more remote 
parts of the earth, that many interesting effects are produced. The 
most noticeable effect is produced upon the ocean, as would be nat- 
urally expected. The tides of the ocean, in their flow and ebb, in- 
dicate this disturbance. The extent of the rise and fall is greater as 
the moon is nearer. It is probable that the tides would be so great 
if the moon was at half its present distance from the earth, that our 



20 THE EARTH AND ITS CHIEF MOTIONS. 

world would be a desolate, uninhabited waste, the immense tide 
waves sweeping over the continents twice a day. 

Other effects even more disastrous would result from bringing 
these bodies too near together. Thus in such immense masses of 
matter, cohesion is a force inferior to gravity ; hence the superior 
gravity of the earth would not only more than counterbalance the 
lunar gravity of the parts next to the earth ; but might even be in 
excess sufficient to break loose from the moon, mountains and other 
vast masses and cause them to fall upon the earth ; and thus the 
moon might be literally torn to pieces, and the earth might be over- 
whelmed by the wreck falling upon it. 

note c. 

When two bodies revolve about their common centre of gravity, 
it is evident that the form of the paths described, (the orbits), will 
d-pend in part upon the nature of the force which urges them to- 
ward each other. Thus when two bodies are attached to the ends 
of a rod, and made to revolve, each body describes a circle about 
the common centre of gravity. If the masses of the two bodies are 
greatly disproportioned, as they are for instance in the case of the Earth 
and Sun, we say the smaller body describes a circle about the larger. 
When two bodies in space form a system, the form of the orbit de- 
scribed will depend upon the intensity of the mutual gravity, and 
upon the direction and velocity of the planet ; when the direction of 
the motion is nearly perpendicular to the line joining the planet and 
the Sun, (the radius vector) ; and the mutual gravity is almost exact- 
ly balanced by centrifugal force; the orbit is nearly a circle. It is an 
ellipse of small eccentricity. This is the case with the earth and 
the other planets. It is conceivable that a planet revolve about 
the sun in an exact circle. 

So the orbit under other conditions may be an ellipse of greater 
eccentricity, and even a parabola ; as is the case with some comets. 
Still another possible form of the orbit is the hyperbola. But in any 
case of a body urged towards the Sun by gravity, whatever be the 
velocity, and whatever be the direction, excepting motion in a direct 
line towards or from the Sun, it will move in an orbit of one or 
another of the kinds named. These curves belong to one class 
called Conic Sections. 



Till-: EARTH AND ITS CHIEF .MOTIONS. 



CHAPTER VIII. 

THK TANGENT INDEX. 

We are now prepared to consider the questions, at what rate is 
the earth moving in its orbit ? and, what is the direction in space of 
the earth's motion at this moment? 

The earth's distance from the sun is about 92,000,000 miles. The 
orbit of the earth, its path, is nearly a circle — it is slightly elliptical. 
Counting it a circle, w T e multiply twice 92,000,000 by 3. 1416 to find 
the circumference of the circle. This product is the distance the 
earth travels each year. It is nearly 580,000,000 miles. As the or- 
bit is elliptical, the earth moves some faster when nearest the sun, 
about January 1st, and a little slower when farthest distant, about 
July 1st. Disregarding these inequalities, which are small, we find 
the earth moves about 1,500,000 miles each day. This gives a veloc- 
ity of about 67,000 miles per hour ; of 1,100 miles per minute, and 
nearly iSyi miles per second. This velocity far exceeds that of the 
swiftest moving objects which usually occur to us in making compar- 
isons. A railway train of cars can be made to move one mile in a 
minute. The earth moves 1,100 times as fast. A cannon-ball may 
have, at the instant of discharge, a velocity of about 2,000 feet per 
second. The earth moves 50 times as fast. An illustration is some- 
times given in this way : Suppose the cannon to be discharged in 
the direction of the earth's motion. While its velocity seems to us 
so great, yet, in fact, there has been added to its velocity, which it 
had in common with the earth, a small percentage — that is, its veloc- 
ity is increased from 100,000 feet to 102,000 feet per second. On 
the other hand, if the cannon be discharged in the opposite direction ; 
while to us it seems to be moving swiftly in that direction, it is really 
still going along in the same direction as the earth, but with a veloc- 
ity relatively very little less — that is, the ball is going in the same 
direction as the earth 98,000 feet per second, while the cannon moves 
along with the earth 100,000 feet per second. Moreover if in this 



22 THE EARTH AND ITS CHIEF MOTIONS. 

last case the direction of discharge takes in some obstruction, as the 
wall of a fort or the side of a ship, at a suitable distance, the ball 
seems to us to dash against the obstacle; but, in fact, the obstacle 
dashes against the ball with a speed equal to their difference in veloc- 
ity — that is 2,000 feet per second. 

In respect to the direction of the earth's motion in its orbit, the 
general statement is that the earth, with all the planets, moves round 
the sun from west to east. But the meaning of these terms, west 
and east, is somewhat different here from their common meaning ; 
for as the earth turns on its axis every twenty four hours, ea^t as we 
generally understand it, is quite a different direction in space at dif- 
ferent times of the day ; also, as the earth pass s on around the sun, 
east in its orbit means very different directions at different times of 
the year. We have, so to speak, to detach our ideas of east and 
west from our horizon; a rather difficult thing: to do, but which the 
tangent index may perhaps assist in doing. 

The earth, in going around the sun, in the course of a year is 
moving in every direction in the plane of the ecliptic — that is, the 
earth changes the direction of its motion in space 360 degrees each 
year, which is nearly one degree per day. But the earth, by turning 
on its axis in one day, turns our horizon in different successive di- 
rections, so that it is puzzling to keep before the mind the true di- 
rection. The index helps us in this by the aid of certain quantities, 
geocentric co-ordinates — namely, the declination of the tangent of 
the earth's orbit, and its hour angle. Declination ha* the same 
meaning as given heretofore. Hour angle, as used here, means the 
diedral or opening between two great circles of the celestial sphere 
— namely, the hour circle or meridian, which contains the tangent, 
and the hour circle of the sun. These quantities vary in value from 
day to day, but their value depends on the place of the earth in its 
orbit. As the earth is in nearly the same place in its orbit on a given 
date year after year, these co-ordinate* can b calculated, and th c ir 
values entered in a table in connection with the corresponding 
dates. The accompanying table gives these values for dates distrib- 
uted through the year at sufficiently small intervals ; also, on the in- 
strument on the arc of declination and of hour angle are placed 
dates at intervals of five degrees, from which, by inspection, a suffi- 



THE EARTH AND ITS CHIEF MOTIONS. 23 

ciently exa6t estimate of the value for intermediate dates can be 
made. 

To use the instrument, place it on a horizontal table, and in the 
meridi n, that is, north and south, with the higher end to the north. 
This can be most conveniently done bv a compass needle. Then 
raise or lower the north end of the axis so that its elevation shall 
equal the latitude of he place. The latitude arc is graduated and 
numbered, so that this is done ea ily. The axis is to be clamped in 
this position. Next, set the tangent to the proper declination, either 
by an inspection of the dates on the arc, or by referring to the table; 
also set the hour angle for the date. Then turn the instrument on 
the axis till the time index points to the true solar time of day. The 
tangent index is now a tangent to the earth's orbit, and it points out 
the direction of the earth's motion in space. Moreover, if the in- 
strument be turned so as to keep up with the time, or if it be turned 
forward from time to time, it will continue to point the same direc- 
tion in space the entire twenty-four hours — that is, the eff ct of the 
earth's diurnal motion is eliminated. 

Notice that by the time of day we mean astronomical time — that 
is, the day begins at noon, and the hours are numbered to twenty- 
four ; therefore to the a. M. hours we add to our watch or clock time 
twelve hours. Also, if we w sh to be very exa6l we take account of 
the difference between mean and apparent time — that i c , we look in 
the almanac at the date, and if the sun is fast a certain number of min- 
utes, we add that number of minutes to the time given by our time- 
piece; or if the sun is slow we subtract If our time is standard 
(rai'road) time, it must first be changed to local time. 

The setting of the declination and hour angle need not be changed 
during the day. Indeed, an inspection of the table will show that 
the change from one day to the next is mall, only about one-third 
of a degree at most. 

Thus at any hour of the day or night, and on any day of the year, 
we may exhibit with this instrument with certainty and with all desir- 
able accuracy, the true direction in space of the earth's motion in 
its orbit. 



24 



THE EARTH AND ITS CHIEF MOTIONS. 




THE TANGENT INDEX. 



A, Latitude arc. 

B, Tangent index. 

C, Tangent declination arc. 

D, Axis of the earth. 

jE, Solar declination arc. 

F, Radius vector. 

G, Time circle. 

H, Time index and hour angle arc. 



THE EARTH AND its CHIEF MOTIONS. 



25 



CHAPTER IX. 

THE TANGENT INDEX TABLE FORMULAS AND DEMONSTRATION. 

Table of Declinations and of Hoar A)igles. 

The declinations are found in column D. The sign for any date 
is found at the head and loot of the column containing the date, 
+ means north declinations, — means south declinations. H is the 
hour angle. There are two columns of H, and each of these has 
two columns of corresponding dates : 



March 20 
March 25 
March 30 
April 4 
April 9 
April 14 
April 19 
April 24 
April 29 
May 4 
May 9 
May 14 
May 19 
May 24 
May 29 
June 3 
June S 
June 13 
June 18, 
June 21 



90 
89 

88 

3; 
86 
86 
85 

85 

§5 
85 

s 5 
35 
86 

86 

87 

8- 



21 
35 
53 

17 

46 
24 
10 

4 
6 

17 

36 

S 

34 
12 



00 

88 42 

89 30 

90 00 



Sept. 23 23° 

Sept. 27 23 

Oct. 3 2\ 

Oct. 8 22 

Oct. 13 22 

Oct. 17 21 

Oct. 22 20 

Oct. 27 19 

Nov. 1 18 

Nov. 6 16 

Nov. n 15 

Nov. 16 13 

Nov. 20 12 

Nov. 25 10 

Nov. 30 8 

Dec. 4 6 

Dec. 9 4 

Dec. 14 3 

Dec. 19 1 

Dec. 21 .... o 



27' March 20 ... 90° 

22 March 15 90 

6 jMarch 10 91 

39 jMarch 5 92 

1 "March 1 93 

14 Feb. 24 93 

19 Feb. 19 94 

13 Feb. 14 94 

o Feb. 9 94 

40 Feb. 4 94 

13 Jan. 31 94 

40 Jan. 26 94 

2 Jan. 21 94 

20 Jan. 16 93 

34 Jan. 12 93 

46 Jan. 7 92 

55 Jan. 2 92 

2 Dec. 29 91 

9 Dec. 24 90 

o Dec. 21 90 



o' Sept. 23. 

49 Sept. 18. 
39 Sept. 13. 

25 Sept. 8. 
7 Sept. 3. 

43 Aug. 28., 

14 Aug. 23. 

36 Aug. 18., 

50 Aug. 13., 
56 Aug. S., 
54 Aug. 3., 
43 July 29., 
24 July 14.. 
52 July 19.. 

26 Julv 14., 
48 Julv 9., 

5 July 4., 

18 June 29., 

30 June 24., 

00 June 2i., 



Formulas for computing the numbers in the table : 

Sin D = — sin e cos 1. 

Cos (180 — H) = tan D tan d. 

D is the tano-ent declination. 



H is the tangent hour angle. 

4 



26 



THE EARTH AND ITS CHIEF MOTIONS. 



e is the obliquity of the ecliptic, 23K degrees, nearly. 

1 is the Sun's longitude. 

d is the Sun's declination. 

e, 1 and d are taken from the Solar tables, for the date, e may be 
considered constant. 

In computing the table above, the Astronomical Ephemeris, for 
1886, was used. The values of 1 and d were taken at intervals of 
five days through the quarter, March 2ist-June 21st, Greenwich 
noon of the date. For the other three quarters of the year, the 
dates were selected with reference to the value of 1. Hence, the 
values of D and H may not be correcl to the minute at Greenwich 
noon ; but will represent the values nearly enough for the purpose. 
As for other years than 1886; while the values in the table are not 
precise, they will be found exa6t enough for the purpose of the 
table for an indefinite number of years. 

DEMONSTRATION OF FORMULAS. 

The formulas above are demonstrated by the aid of the accom- 
panying diagram: 




THE EARTH AND its CHIEF Morio.N'S. 27 

This represents the celestial sphere. E is the Earth's centre. 
N E is the axis. N is the north pole. A D C is the equinoctial. 
A B C is the ecliptic, the Sun's apparent annual path. A is the 
vernal equinox. The angle K A D is the obliquity of the ecliptic, 
represented in the formula by e. Let S be the place of the Sun at 
some time. Then the arc A S is the Sun's longitude, represented 
by 1. Draw S E: this is the radius vector of the orbit. Draw E P 
in the plane of the ecliptic perpendicular to S E. Then is E P 
the tangent to the Earth's orbit. The orbit is not represented 
in the diagram. As the Sun's apparent motion is towards C, the 
Earth's real motion is in the direction E P. Draw the hour circles 
N S T and NPQ. S T is the Sun's declination, represented by d, 
and P Q is the declination of the tangent, represented by D. Also 
the angle PXS which equals the arc Q T, is the hour angle of the 
tangent, represented by H. The arc P S is a quadrant, 90 degrees. 

In the spherical triangle P A O, Q a right angle, Sin P O = sin 
A P sin P A Q (1). But P A = A S — P S = 1 — 90 = — (90 — 1). 
Therefore, Sin P A = — sin (90 — 1) = — cos 1. Substituting in 
(1) for sin P A this value, and for P Q and P A Q their symbols, we 
have Sin D = — sin e cos 1. 

Again, in the quadrantal triangle P N S, PN = 90 — D and N S 
= 90° — d. It is convenient to solve the triangle polar to P N S, 
and therefore right angled. The hypothenuse of this triangle is 
180 — P N S or 180 — H. The adjacent angles are 180 — P N 
and 180 — N S. That is 180 — (90 — D) = 90 + D for the first ; 
and 180 — (90 — d) = 90 + d for the second. Then cos (180 — H) 
= cot (90 + D) cot (90 -I- d) = (— tan D) (— tan d) = tan D tan d. 

We may also find H = Q T by first computing A O in the triangle 
P A O. Then OT = AT-AO. A T is the Sun's right ascen- 
sion. Calling this a, and calling A O, the right ascension of the 
tangent, A, we have H = a — A. The formula for A is obtained 

., ^ A ~ ran A P tan (A S — P S) ^, ... * tan (1 - 90 

thus: Tan AQ = - P A q = — ; osPAQ . That is tan A = cose — 

— tan (90° — 1) cot 1 

cos e cos e 

It can be seen, either from the formula, or from the diagram, that 
D varies in value between the limits — e and -f- e ; the former l imit 
being the value of D when the Sun is at the vernal equinox ; the 



28 THE EARTH AND ITS CHIEF MOTIONS. 

latter being the value of D when the Sun is at the autumnal equi- 
nox. Its value is o° when the Sun is at either solstice. 

Also, it is seen that H has the value 90 when D is o°, which is the 
case as stated above, when the Sun is at either solstice ; or when d 
is o°, which is the case when the Sun is at either equinox. About 
midway between the times for these values, H has a minimum value 
of 85 4/, viz., about May 4th and November 6th ; and a maximum 
value of 94 56', viz., about February 4th and August 8th. These 
facts can also be seen from an inspection of the table. 

The analysis above is based upon the cordition that the tangent 
to the Earth's orbit, and the radius vector to the point of contact 
are perpendicular to each other ; that is, that the orbit is a circle 
with the Sun at the center. But the tangent is perpendicular to the 
radius vector only at perihelion and aphelion ; that is, about January 
1 st and July 1st. The formulas therefore are strictly true only for 
these times, and only approximately true for other times. But on 
account of the small eccentricity of the orbit, the angle between the 
tangent and radius vector differs from 90 less than a degree at the 
most. Thus, on April 1st, when the Earth is at its mean distance 
from the Sun, the forward end of the tangent makes with the radius 
vector an angle of 90 5s 7 nearly. And, again, about October 2nd 
the angle is about 89 2 7 . These are the extreme values. If the 
exact value of this angle were used for the several dates, in com- 
puting the values of D and H, these would be found to differ from 
those given in the table only a few minutes in any case. The num- 
bers in the table may therefore be considered sufficiently exact for 
use in connection with the Tangent Index. 

The discussion of the astronomical relations of the Earth, accord- 
ing to the scheme proposed at the beginning, is now completed. 
The presentation of the general plan of the Solar System was not in 
the scheme ; neither the consideration of the perturbations, the ine- 
qualities in motion caused by the mutual gravity of the several 
members of the system. These effects are suitably discussed in 
many excellent works on Astronomy, to which the student is referred 
for information on these subjects. The end proposed in this discus- 
sion is accomplished if the form and principal motions of the Earth 
are more clearly seen. 



LIBRARY OF CONGRESS 



003 538 886 2 




THE TANGENT INDEX. 

A Device of the Author. 



